The Well-Tempered Clavier …what does it actually imply?In this episode I seek (in 18 minutes) to demonstrate mathematically pure intervals, alongside ‘tempered’ intervals. The circle of fifths is in fact a spiral of fifths— it is infinite. We seek to make it a circle for convenience, but this means that the distance between what would be mathematically pure intervals must be altered (!) in order to force the spiral into a circle. The question remains: how do we do that? Music today is heard i...
The Well-Tempered Clavier …what does it actually imply?
In this episode I seek (in 18 minutes) to demonstrate mathematically pure intervals, alongside ‘tempered’ intervals. The circle of fifths is in fact a spiral of fifths— it is infinite. We seek to make it a circle for convenience, but this means that the distance between what would be mathematically pure intervals must be altered (!) in order to force the spiral into a circle.
The question remains: how do we do that? Music today is heard in equal temperament, where all keys, all tonalities sound the same. Pure intervals have been forced into identical ‘impure’ ones— convenient, but this deprives us not only of a beautiful natural simplicity, but also the individual character of each key found in unequal temperaments. From the earliest European music, numerous solutions to this immortal problem have been offered. By the time Bach arrives, his solution, laid out for us in The Well-Tempered Clavier, is certainly an elegant one…but we don’t quite know exactly what it is.
The remainder of the episode analyzes the prelude and fugue, no. 13, in F-sharp major, BWV 858. A look at the opening of the prelude in earliest version offers lovely insights into Bach’s working mind:
In revision Bach changes the repeated note — a recognizable motif of the early version— into a trill, now acting as its own motif:
Particularly fascinating is the way Bach changes the harmonic rhythm from the early version here, bar 17:
Now again from measure 17 in the revision. Bach inserts two full bars, extending the harmonic length of g# minor and C# Major:
“Thank Bach for God.”
A huge thanks to Bradly Lehman for helping preparing this episode. I haven’t even scratched the surface of what his work covers, but hopefully you’ve got some idea how deep the topic of tuning can go. Lehman has some great online resources illuminating the fascinating world of temperament. For starters, try:
www.larips.com (Spiral spelled backward)
Dig deeper with this essay here, published by the Reimenschneider Bach Institute:
And finally, more Articles and Essays by Bradley Lehman
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Concepts Covered:
Pure vs tempered intervals, ‘spiral of fifths,’ vs. the circle of fifths, equal temperament, loss of key character, Bradly Lehman temperament, historical tuning systems, Bach’s Well-Tempered Clavier, temperament and tuning, F-sharp major Prelude and Fugue No. 13, Bach’s compositional revisions, and BWV 858
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